Tài liệu Đề tài " Isomonodromy transformations of linear systems of difference equations" pptx

Annals of Mathematics

Isomonodromy
transformations of linear
systems of difference
equations

By Alexei Borodin


Annals of Mathematics, 160 (2004), 1141–1182

Isomonodromy transformations
of linear systems of difference equations
By Alexei Borodin

Abstract
We introduce and study “isomonodromy” transformations of the matrix
linear difference equation Y (z + 1) = A(z)Y (z) with polynomial A(z). Our
main result is construction of an isomonodromy action of Zm(n+1)−1 on the
space of coefficients A(z) (here m is the size of matrices and n is the degree of
A(z)). The (birational) action of certain rank n subgroups can be described by
difference analogs of the classical Schlesinger equations, and we prove that for
generic initial conditions these difference Schlesinger equations have a unique
solution. We also show that both the classical Schlesinger equations and the
Schlesinger transformations known in isomonodromy theory, can be obtained
as limits of our action in two different limit regimes.
Similarly to the continuous case, for m = n = 2 the difference Schlesinger
equations and their q-analogs yield discrete Painlev´ equations; examples ine
clude dPII, dPIV, dPV, and q-PVI.
Introduction

In recent years there has been considerable interest in analyzing a certain
class of discrete probabilistic models which in appropriate limits converge to
well-known models of random matrix theory. The sources of these models are
quite diverse, they include combinatorics, representation theory, percolation
theory, random growth processes, tiling models and others.
One quantity of interest in both discrete models and their random matrix
limits is the gap probability – the probability of having no particles in a given
set. It is known, due to works of many people (see [JMMS], [Me], [TW],
[P], [HI], [BD]), that in the continuous (random matrix type) setup these
probabilities can be expressed through solution of an associated isomonodromy
problem for a linear system of differential equations with rational coefficients.
The goal of this paper is to develop a general theory of “isomonodromy”
transformations for linear systems of difference equations with rational coefficients. This subject is of interest in its own right. As an application of


1142

ALEXEI BORODIN

the theory, we show in a subsequent publication that the gap probabilities
in the discrete models mentioned above are expressible through solutions of
isomonodromy problems for such systems of difference equations. In the case
of one-interval gap probability this has been done (in a different language) in
[Bor], [BB]. One example of the probabilistic models in question can be found
at the end of this introduction.
Consider a matrix linear difference equation
(1)

Y (z + 1) = A(z)Y (z).

Here
A(z) = A0 z n + A1 z n−1 + · · · + An ,

Ai ∈ Mat(m, C),

is a matrix polynomial and Y : C → Mat(m, C) is a matrix meromorphic
function. 1 We assume that the eigenvalues of A0 are nonzero and that their
ratios are not real. Then, without loss of generality, we may assume that A0
is diagonal.
It is a fundamental result proved by Birkhoff in 1911, that the equation (1) has two canonical meromorphic solutions Y l (z) and Y r (z), which are
holomorphic and invertible for z
0 and z
0 respectively, and whose
asymptotics at z = ∞ in any left (right) half-plane has a certain form. Birkhoff
further showed that the ratio
P (z) = (Y r (z))−1 Y l (z),
which must be periodic for obvious reasons, is, in fact, a rational function in
exp(2πiz). This rational function has just as many constants involved as there
are matrix elements in A1 , . . . , An . Let us call P (z) the monodromy matrix of
(1).
Other results of Birkhoff show that for any periodic matrix P of a specific
form, there exists an equation of the form (1) with prescribed A0 , which has
P as the monodromy matrix. Furthermore, if two equations with coefficients
A(z) and A(z), A0 = A0 , have the same monodromy matrix, then there exists
a rational matrix R(z) such that
(2)

A(z) = R(z + 1)A(z)R−1 (z).

The first result of this paper is a construction, for generic A(z), of a homomorphism of Zm(n+1)−1 into the group of invertible rational matrix functions,

such that the transformation (2) for any R(z) in the image, does not change
the monodromy matrix.
If we denote by a1 , . . . , amn the roots of the equation det A(z) = 0 (called
eigenvalues of A(z)) and by d1 , . . . , dn certain uniquely defined exponents of
the asymptotic behavior of a canonical solution Y (z) of (1) at z = ∞, then
1

Changing Y (z) to (Γ(z))k Y (z) readily reduces a rational A(z) to a polynomial one.


ISOMONODROMY TRANSFORMATIONS

1143

the action of Zm(n+1)−1 is uniquely defined by integral shifts of {ai } and {dj }
with the total sum of all shifts equal to zero. (We assume that ai − aj ∈ Z and
/
/
di − dj ∈ Z for any i = j.)
The matrices R(z) depend rationally on the matrix elements of {Ai }n
i=1
and {ai }mn (A0 is always invariant), and define birational transformations of
i=1
the varieties of {Ai } with given {ai } and {dj }.
There exist remarkable subgroups Zn ⊂ Zm(n+1)−1 which define birational
transformations on the space of all A(z) (with fixed A0 and with no restrictions on the roots of det A(z)), but to see this we need to parametrize A(z)
differently.
To define the new coordinates, we split the eigenvalues of A(z) into n
groups of m numbers each:
(1)

(n)

{a1 , . . . , amn } = {a1 , . . . , a(1) } ∪ · · · ∪ {a1 , . . . , a(n) }.
m
m
The splitting may be arbitrary. Then we define Bi to be the uniquely determined (remember, everything is generic) element of Mat(m, C) with eigenvalues

(i) m

aj

j=1

, such that z − Bi is a right divisor of A(z):
A(z) = (A0 z n−1 + A1 z n−1 + · · · + An−1 )(z − Bi ).

The matrix elements of {Bi }n are the new coordinates on the space of A(z).
i=1
The action of the subgroup Zn mentioned above consists of shifting the
eigenvalues in any group by the same integer assigned to this group, and also
shifting the exponents {di } by the same integer (which is equal to minus the
sum of the group shifts). If we denote by {Bi (k1 , . . . , kn )} the result of applying
k ∈ Zn to {Bi }, then the following equations are satisfied:
(3)

Bi (. . . ) − Bi (. . . , kj + 1, . . . ) = Bj (. . . ) − Bj (. . . , ki + 1, . . . ),

(4)

Bj (. . . , ki + 1, . . . )Bi (. . . ) = Bi (. . . , kj + 1, . . . )Bj (. . . ),

(5)

Bi (k1 + 1, . . . , kn + 1) = A−1 Bi (k1 , . . . , kn )A0 − I,
0

where i, j = 1, . . . , n, and dots in the arguments mean that other kl ’s remain
unchanged. We call them the difference Schlesinger equations for the reasons
that will be clarified below. Note that (3) and (4) can be rewritten as
z − Bi (. . . , kj + 1, . . . ) z − Bj (. . . ) = z − Bj (. . . , ki + 1, . . . ) z − Bi (. . . ) .
Independently of Birkhoff’s general theory, we prove that the difference
Schlesinger equations have a unique solution satisfying
(6)

Sp(Bi (k1 , . . . , kn )) = Sp(Bi ) − ki ,

i = 1, . . . , n,

for an arbitrary nondegenerate A0 and generic initial conditions {Bi = Bi (0)}.
(The notation means that the eigenvalues of Bi (k) are equal to those of Bi
shifted by −ki .) Moreover, the matrix elements of this solution are rational


1144

ALEXEI BORODIN

functions in the matrix elements of the initial conditions. This is our second
result.

In order to prove this claim, we introduce yet another set of coordinates
on A(z) with fixed A0 , which is related to {Bi } by a birational transformation.
It consists of matrices Ci ∈ Mat(m, C) with Sp(Ci ) = Sp(Bi ) such that
A(z) = A0 (z − C1 ) · · · (z − Cn ).
In these coordinates, the action of Zn is described by the relations
(7)

z + 1 − Ci · · · z + 1 − Cn A0 z − C1 · · · z − Ci−1
= z + 1 − Ci+1 · · · z + 1 − Cn A0 z − C1 · · · z − Ci ,

Cj = Cj (k1 , . . . , kn ),

Cj = Cj (k1 , . . . , ki−1 , ki + 1, ki+1 , . . . , kn ) for all j.

Again, we prove that there exists a unique solution to these equations satisfying Sp(Ci (k)) = Sp(Ci ) − ki , for an arbitrary invertible A0 and generic
{Ci = Ci (0)}. The solution is rational in the matrix elements of the initial
conditions.
The difference Schlesinger equations have an autonomous limit which consists of (3), (4), and
(5-aut)
(6-aut)

Bi (k1 + 1, . . . , kn + 1) = A−1 Bi (k1 , . . . , kn )A0 ,
0
Sp(Bi (k1 , . . . , kn )) = Sp(Bi ),

i = 1, . . . , n.

The equation (7) then becomes
(7-aut)

z − Ci · · · z − Cn A0 z − C1 · · · z − Ci−1
= z − Ci+1 · · · z − Cn A0 z − C1 · · · z − Ci .

The solutions of these equations were essentially obtained in [V] via a general construction of commuting flows associated with set-theoretical solutions
of the quantum Yang-Baxter equation; see [V] for details and references.
The autonomous equations can also be explicitly solved in terms of abelian
functions associated with the spectral curve {(z, w) : det(A(z) − wI) = 0},2
very much in the spirit of [MV, §1.5]. We hope to explain the details in a
separate publication.
The whole subject bears a strong similarity (and not just by name!) to the
theory of isomonodromy deformations of linear systems of differential equations
with rational coefficients:
(8)

2

dY(ζ)
=
dζ

n

B∞ +
k=1

Bi
ζ − xi

Y(ζ),

It is easy to see that the curve is invariant under the flows.


ISOMONODROMY TRANSFORMATIONS

1145

which was developed by Schlesinger around 1912 and generalized by Jimbo,
Miwa, and Ueno in [JMU], [JM] to the case of higher order singularities. If
we analytically continue any fixed (say, normalized at a given point) solution
Y(ζ) of (8) along a closed path γ in C avoiding the singular points {xk } then
the columns of Y will change into their linear combinations: Y → YMγ . Here
Mγ is a constant invertible matrix which depends only on the homotopy class
of γ. It is called the monodromy matrix corresponding to γ. The monodromy
matrices define a linear representation of the fundamental group of C with
n punctures. The basic isomonodromy problem is to change the differential
equation (8) so that the monodromy representation remains invariant.
There exist isomonodromy deformations of two types: continuous ones,
when xi move in the complex plane and Bi = Bi (x) form a solution of a system of partial differential equations called Schlesinger equations, and discrete
ones (called Schlesinger transformations), which shift the eigenvalues of Bi and
exponents of Y(ζ) at ζ = ∞ by integers with the total sum of shifts equal to 0.
We prove that in the limit when
Bi = xi ε−1 + Bi ,

ε → 0,

our action of Zm(n+1)−1 in the discrete case converges to the action of Schlesinger
transformations on Bi . This is our third result.
Furthermore, we argue that the “long-time” asymptotics of the Zn -action
in the discrete case (that is, the asymptotics of Bi ([x1 ε−1 ], . . . , [xn ε−1 ])),

ε small, is described by the corresponding solution of the Schlesinger equations. More exactly, we conjecture that the following is true.
Take Bi = Bi (ε) ∈ Mat(m, C), i = 1, . . . , n, such that
Bi (ε) − yi ε−1 + Bi → 0,

ε → 0.

Let Bi (k1 , . . . , kn ) be the solution of the difference Schlesinger equations (3.1)–
(3.3) with the initial conditions {Bi (0) = Bi }, and let Bi (x1 , . . . , xn ) be the
solution of the classical Schlesinger equations (5.4) with the initial conditions
{Bi (y1 , . . . , yn ) = Bi }. Then for any x1 , . . . , xn ∈ R and i = 1, . . . , n, we have
Bi [x1 ε−1 ], . . . , [xn ε−1 ] +[xi ε−1 ]−yi ε−1 +Bi (y1 −x1 , . . . , yn −xn ) → 0, ε → 0.
In support of this conjecture, we explicitly show that the difference
Schlesinger equations converge to the conventional Schlesinger equations in
the limit ε → 0.
Note that the monodromy representation of π1 (C \ {x1 , . . . , xn }) which
provides the integrals of motion for the Schlesinger flows, has no obvious analog
in the discrete situation. On the other hand, the obvious differential analog
of the periodic matrix P , which contains all integrals of motion in the case
of difference equations, gives only the monodromy information at infinity and
does not carry any information about local monodromies around the poles
x1 , . . . , xn .


1146

ALEXEI BORODIN

Most of the results of the present paper can be carried over to the case
of q-difference equations of the form Y (qz) = A(z)Y (z). The q-difference
Schlesinger equations are, cf. (3)–(6),

(3q)

Bi (. . . ) − Bi (. . . , q kj +1 , . . . ) = Bj (. . . ) − Bj (. . . , q ki +1 , . . . ),

(4q)

Bj (. . . , q ki +1 , . . . )Bi (. . . ) = Bi (. . . , q kj +1 , . . . )Bj (. . . ),

(5q)

Bi (q k1 +1 , . . . , q kn +1 ) = q −1 A−1 Bi (q k1 , . . . , q kn )A0 ,
0

(6q)

Sp(Bi (q k1 , . . . , q kn )) = q −ki Sp(Bi ),

i = 1, . . . , n.

The q-analog of (7) takes the form
(7q)

z − q −1 Ci · · · z − q −1 Cn A0 z − C1 · · · z − Ci−1
= z − q −1 Ci+1 · · · z − q −1 Cn A0 z − C1 · · · z − Ci ,

Cj = Cj (q k1 , . . . , q kn ),

Cj = Cj (q k1 , . . . , q ki−1 , q ki +1 , q ki+1 , . . . , q kn ) for all j.

A more detailed exposition of the q-difference case will appear elsewhere.

Similarly to the classical case, see [JM], discrete Painlev´ equations of
e
[JS], [Sak] can be obtained as reductions of the difference and q-difference
Schlesinger equations when both m (the size of matrices) and n (the degree
of the polynomial A(z)) are equal to two. For examples of such reductions
see [Bor, §3] for difference Painlev´ II equation (dPII), [Bor, §6] and [BB, §9]
e
for dPIV and dPV, and [BB, §10] for q-PVI. This subject still remains to be
thoroughly studied.
As was mentioned before, the difference and q-difference Schlesinger equations can be used to compute the gap probabilities for certain probabilistic
models. We conclude this introduction by giving an example of such a model.
We define the Hahn orthogonal polynomial ensemble as a probability measure
on all l-point subsets of {0, 1, . . . , N }, N > l > 0, such that
l

Prob{(x1 , . . . , xl )} = const ·

(xi − xj )2 ·
1≤i
w(xi ),
i=1

where w(x) is the weight function for the classical Hahn orthogonal polynomials:
α+x β+N −x
w(x) =
,
α, β > −1 or α, β < −N.
x
N −x

This ensemble came up recently in harmonic analysis on the infinite-dimensional
unitary group [BO, §11] and in a statistical description of tilings of a hexagon
by rhombi [Joh, §4].
The quantity of interest is the probability that the point configuration
(x1 , . . . , xl ) does not intersect a disjoint union of intervals [k1 , k2 ] · · ·
[k2s−1 , k2s ]. As a function in the endpoints k1 , . . . , k2s ∈ {0, 1, . . . , N }; this


1147

ISOMONODROMY TRANSFORMATIONS

probability can be expressed through a solution of the difference Schlesinger
equations (3)–(6) for 2 × 2 matrices with n = deg A(z) = s + 2, A0 = I,
Sp(Bi ) = {−ki , −ki },
Sp(B2s+1 )

i = 1, . . . , 2s,

Sp(B2s+2 ) = {0, −α, N + 1, N + 1 + β},

and with certain explicit initial conditions. The equations are also suitable for
numerical computations, and we refer to [BB, §12] for examples of those in the
case of a one interval gap.
I am very grateful to P. Deift, P. Deligne, B. Dubrovin, A. Its, D. Kazhdan,
I. Krichever, G. Olshanski, V. Retakh, and A. Veselov for interesting and
helpful discussions.
This research was partially conducted during the period the author served
as a Clay Mathematics Institute Long-Term Prize Fellow.
1. Birkhoff ’s theory

Consider a matrix linear difference equation of the first order
(1.1)

Y (z + 1) = A(z)Y (z).

Here A : C → Mat(m, C) is a rational function (i.e., all matrix elements of
A(z) are rational functions of z) and m ≥ 1. We are interested in matrix
meromorphic solutions Y : C → Mat(m, C) of this equation.
Let n be the order of the pole of A(z) at infinity, that is,
A(z) = A0 z n + A1 z n−1 + lower order terms .
We assume that (1.1) has a formal solution of the form
(1.2)

Y (z) = z nz e−nz

ˆ
ˆ
Y1 Y2
ˆ
+ 2 + ...
Y0 +
z
z

diag ρz z d1 , . . . , ρz z dm
1
m

ˆ
with ρ1 , . . . , ρm = 0 and det Y0 = 0.3

It is easy to see that if such a formal solution exists then ρ1 , . . . , ρm must
ˆ
be the eigenvalues of A0 , and the columns of Y0 must be the corresponding
eigenvectors of A0 .
Note that for any invertible T ∈ Mat(m, C), (T Y )(z) solves the equation
(T Y )(z + 1) = (T A(z)T −1 ) (T Y )(z).
Thus, if A0 is diagonalizable, we may assume that it is diagonal without loss
of generality. Similarly, if A0 = I and A1 is diagonalizable, we may assume
that A1 is diagonal.
3

Substituting (1.2) in (1.1) we use the expansion
to compare the two sides.

z+1 nz
z

= enz ln(1+z

−1

)

n

= en − ne + . . .
2z


1148

ALEXEI BORODIN

Proposition 1.1. If A0 = diag(ρ1 , . . . , ρm ), where {ρi }m are nonzero
i=1
and pairwise distinct, then there exists a unique formal solution of (1.1) of the
ˆ
form (1.2) with Y0 = I.
Proof. It suffices to consider the case n = 0; the general case is reduced
to it by considering (Γ(z))n Y (z) instead of Y (z), because
Γ(z) =

√

2π z z− 2 e−z 1 +
1

1 −1
z + ...
12

.

(More precisely, this expression formally solves Γ(z + 1) = zΓ(z).)
Thus, we assume n = 0. Then we substitute (1.2) into (1.1) and compute
ˆ
ˆ
Yk one by one by equating the coefficients of z −l , l = 0, 1, . . . . If Y0 = I then
the constant coefficients of both sides are trivially equal. The coefficients of
z −1 give

(1.3)

ˆ
ˆ
Y1 A0 + diag(ρ1 d1 , . . . , ρm dm ) = A0 Y1 + A1 .

ˆ
This equality uniquely determines {di } and the off-diagonal entries of Y1 , because
ˆ
ˆ
[Y1 , A0 ]ij = (ρj − ρi )(Y1 )ij .
Comparing the coefficients of z −2 we obtain
ˆ
ˆ
ˆ
ˆ
ˆ
(Y2 − Y1 )A0 + Y1 diag(ρ1 d1 , . . . , ρm dm ) + . . . = A0 Y2 + A1 Y1 + . . . ,
where the dots stand for the terms which we already know (that is, those
ˆ
which depend only on ρi ’s, di ’s, Ai ’s, and Y0 = I). Since the diagonal values of
A1 are exactly ρ1 d1 , . . . ρn dn by (1.3), we see that we can uniquely determine
ˆ
ˆ
the diagonal elements of Y1 and the off-diagonal elements of Y2 from the last
equality.
ˆ
ˆ
Now let us assume that we already determined Y1 , . . . , Yl−2 and the offˆl−1 by satisfying (1.1) up to order l − 1. Then comparing
diagonal entries of Y

the coefficients of z −l we obtain
ˆ
ˆ
ˆ
ˆ
ˆ
(Yl − (l − 1)Yl−1 )A0 + Yl−1 diag(ρ1 d1 , . . . , ρm dm ) + . . . = A0 Yl + A1 Yl−1 + . . . ,
where the dots denote the terms depending only on ρi ’s, di ’s, Ai ’s, and
ˆ
ˆ
Y0 , . . . , Yl−2 . This equality allows us to compute the diagonal entries of Yl−1
and the off-diagonal entries of Yl . Induction on l completes the proof.
The condition that the eigenvalues of A0 are distinct is not necessary for
the existence of the asymptotic solution, as our next proposition shows.
Proposition 1.2. Assume that A0 = I and A1 = diag(r1 , . . . , rn ) where
/
ri − rj ∈ {±1, ±2, . . . } for all i, j = 1, . . . , n. Then there exists a unique formal
ˆ
solution of (1.1) of the form (1.2) with Y0 = I.


ISOMONODROMY TRANSFORMATIONS

1149

Proof. As in the proof of Proposition 1.1, we may assume that n = 0.
Comparing constant coefficients we see that ρ1 = · · · = ρm = 1. Then equating
the coefficients of z −1 we find that di = ri , i = 1, . . . , m. Furthermore, equating
the coefficients of z −l , l ≥ 2 we find that
ˆ

ˆ
[Yl−1 , A1 ] − (l − 1)Yl−1
ˆ
ˆ
is expressible in terms of Ai ’s and Y1 , . . . , Yl−2 . This allows us to compute all
ˆ
Yi ’s recursively.
We call two complex numbers z1 and z2 congruent if z1 − z2 ∈ Z.
Theorem 1.3 (G. D. Birkhoff [Bi1, Th. III]). Assume that
A0 = diag(ρ1 , . . . , ρm ),
ρi = 0,

i = 1, . . . , m,

ρi /ρj ∈ R for all i = j.
/

Then there exist unique solutions Y l (z) (Y r (z)) of (1.1) such that:
(a) The function Y l (z) (Y r (z)) is analytic throughout the complex plane except possibly for poles to the right (left ) of and congruent to the poles of
A(z) (respectively, A−1 (z − 1));
(b) In any left (right ) half-plane Y l (z) (Y r (z)) is asymptotically represented
by the right-hand side of (1.2).
Remark 1.4. Part (b) of the theorem means that for any k = 0, 1, . . . ,
ˆ
Y l,r (z) z −nz enz diag(ρ−z z −d1 , . . . , ρ−z z −dm ) − Y0 −
m
1

ˆ
ˆ

const
Y1
Yk−1
− · · · − k−1 ≤ k
z
z
z

for large |z| in the corresponding domain.
Theorem 1.3 holds for any (fixed) choices of branches of ln(z) in the left
and right half-planes for evaluating z −nz = e−nz ln(z) and z −dk = e−dk ln(z) , and
of a branch of ln(ρ) with a cut not passing through ρ1 , . . . , ρm for evaluating
ρ−z = e−z ln ρk . Changing these branches yields the multiplication of Y l,r (z)
k
by a diagonal periodic matrix on the right.
Remark 1.5. Birkhoff states Theorem 1.3 under a more general assumption: he only assumes that the equation (1.1) has a formal solution of the form
(1.2). However, as pointed out by P. Deligne, Birkhoff’s proof has a flaw in
case one of the ratios ρi /ρj is real. The following counterexample was kindly
communicated to me by Professor Deligne.
Consider the equation (1.1) with m = 2 and
A(z) =

1 1/z
.
0 1/e


1150

ALEXEI BORODIN

The formal solution (1.2) has the form
Y (z) =

I+

0 a −1
z + ...
0 0

1 0
0 e−z

with a = e/(1 − e).
Actual solutions that we care about have the form
1 u(z)
Y (z) =
0 e−z
where u(z) is a solution of u(z + 1) = u(z) + e−z /z. In a right half-plane we
can take
∞ −(z+n)
e
ur (z) = −
.
z+n
The first order approximation of
gers is

n=0
r (z) anywhere

u
∞

u (z) ∼ −
r

n=0

except near nonpositive inte-

e−(z+n)
ae−z
=
.
z
z

Next, terms can be obtained by expanding 1/(z + n).
In order to obtain a solution which behaves well on the left, it suffices to
cancel the poles:
2πi
ul (z) = ur (z) + 2πiz
.
e
−1
The corresponding solution Y l (z) has the needed asymptotics in sectors of the
form π/2 + ε < arg z < 3π/2 + ε, but it has the wrong asymptotic behavior as
z → +i∞. Indeed, limz→+i∞ ul (z) = −2πi.
On the other hand, we can take
2πi e2πiz

,
e2πiz − 1
which has the correct asymptotic behavior in π/2 − ε < arg z < 3π/2 − ε, but
fails to have the needed asymptotics at −i∞.
ul (z) = ul (z) + 2πi = ur (z) +

Remark 1.6. In the case when |ρ1 | > |ρ2 | > · · · > |ρm | > 0, a result
similar to Theorem 1.3 was independently proved by R. D. Carmichael [C].
He considered the asymptotics of solutions along lines parallel to the real axis
only. Birkhoff also referred to [N] and [G] where similar results had been proved
somewhat earlier.
Now let us restrict ourselves to the case when A(z) is a polynomial in z.
The general case of rational A(z) is reduced to the polynomial case by the
following transformation. If (z − x1 ) · · · (z − xs ) is the common denominator
of {Akl (z)} (the matrix elements of A(z)), then
¯
Y (z) = Γ(z − x1 ) · · · Γ(z − xs ) · Y (z)


1151

ISOMONODROMY TRANSFORMATIONS

¯
¯ ¯
solves Y (z + 1) = A(z)Y (z) with polynomial
¯
A(z) = (z − x1 ) · · · (z − xs )A(z).
Note that the ratio P (z) = (Y r (z))−1 Y l (z) is a periodic function. (The
relation P (z + 1) = P (z) immediately follows from the fact that Y l,r solves

(1.1).) From now on let us fix the branches of ln(z) in the left and right halfplanes mentioned in Remark 1.4 so that they coincide in the upper half-plane.
Then the structure of P (z) can be described more precisely.
Theorem 1.7 ([Bi1, Th. IV]). With the assumptions of Theorem 1.3,
the matrix elements pkl (z) of the periodic matrix P (z) = (Y r (z))−1 Y l (z) have
the form
(1)

(n−1) 2π(n−1)iz

pkk (z) = 1 + ckk e2πiz + · · · + ckk
(0)

(1)

e

+ e2πidk e2πniz ,

(n−1) 2π(n−1)iz

pkl (z) = e2πλkl z ckl + ckl e2πiz + · · · + ckl

e

(k = l),

(s)

where ckl are some constants, and λkl denotes the least integer as great as the
real part of (ln(ρl ) − ln(ρk ))/2πi.

Thus, starting with a matrix polynomial A(z) = A0 z n + A1 z n−1 + · · · + An
with nondegenerate A0 = diag(ρ1 , . . . , ρm ), ρk = ρl for k = l, we construct the
(s)
characteristic constants {dk }, {ckl } using Proposition 1.1 and Theorems 1.3,
1.7.
Note that the total number of characteristic constants is exactly the same
as the number of matrix elements in matrices A1 , . . . , An . Thus, it is natural
to ask whether the map
(s)

(A1 , . . . , An ) → {dk }, {ckl }
is injective or surjective (the constants ρ1 , . . . , ρn being fixed). The following
partial results are available.
Theorem 1.8 ([Bi2, §17]). For any nonzero ρ1 , . . . , ρm , ρi /ρj ∈ R for
/
i = j, there exist matrices A1 , . . . , An such that the equation (1.1) with A0 =
diag(ρ1 , . . . ρm ) either possesses the prescribed characteristic constants {dk },
(s)

(s)

{ckl }, or else constants {dk + lk }, {ckl }, where l1 , . . . , lm are integers.
Theorem 1.9 ([Bi1, Th. VII]). Assume there are two matrix polynomials A (z) = A0 z n + · · · + An and A (z) = A0 z n + · · · + An with
A0 = A0 = diag(ρ1 , . . . ρm ),

ρk = 0,

ρk /ρl ∈ R for k = l,
/

such that the sets of the characteristic constants for the equations Y (z + 1) =
A (z)Y (z) and Y (z + 1) = A (z)Y (z) are equal. Then there exists a rational
matrix R(z) such that
A (z) = R(z + 1)A (z)R−1 (z),


1152

ALEXEI BORODIN

and the left and right canonical solutions Y l,r of the second equation can be
obtained from those of the first equation by multiplication by R on the left:
(Y )l,r = R (Y )l,r .
2. Isomonodromy transformations
The goal of this section is to construct explicitly, for given A(z), rational matrices R(z) such that the transformation A(z) → R(z + 1)A(z)R−1 (z),
cf. Theorem 1.9 above, preserves the characteristic constants (more generally,
(s)
preserves {ckl } and shifts dk ’s by integers).
Let A(z) be a matrix polynomial of degree n ≥ 1, A0 = diag(ρ1 , . . . , ρm ),
and ρi ’s are nonzero and their ratios are not real. Fix mn complex numbers
/
a1 , . . . , amn such that ai − aj ∈ Z for any i = j. Denote by M(a1 , . . . , amn ;
d1 , . . . , dm ) the algebraic variety of all n-tuples of m by m matrices A1 , . . . , An
such that the scalar polynomial
det A(z) = det(A0 z n + A1 z n−1 + · · · + An )
of degree mn has roots a1 , . . . , amn , and ρi di −
the analog of (1.3) for arbitrary n).

n
2

= (A1 )ii (this comes from

Theorem 2.1. For any κ1 , . . . , κmn ∈ Z, δ1 , . . . , δm ∈ Z,
mn

m

κi +
i=1

δj = 0,
j=1

there exists a nonempty Zariski open subset A of M(a1 , . . . , amn ; d1 , . . . , dm )
such that for any (A1 , . . . , An ) ∈ A there exists a unique rational matrix R(z)
with the following properties:
A(z) = R(z + 1)A(z)R−1 (z) = A0 z n + A1 z n−1 + · · · + An ,

A0 = A0 ,

(A1 , . . . , An ) ∈ M(a1 + κ1 , . . . , amn + κmn ; d1 + δ1 , . . . , dm + δm ),
and the left and right canonical solutions of Y (z + 1) = A(z)Y (z) have the
form
Y l,r = R Y l,r ,
where Y l,r are left and right canonical solutions of Y (z + 1) = A(z)Y (z).
The map (A1 , . . . , An ) → (A1 , . . . , An ) is a birational map of algebraic
varieties.
(s)

Remark 2.2. The theorem implies that the characteristic constants {ckl }
for the difference equations with coefficients A and A are the same, while the
constants dk are being shifted by δk ∈ Z.


1153

ISOMONODROMY TRANSFORMATIONS

Note also that if we require that all dk ’s do not change, then, by virtue of
Theorem 1.9, Theorem 2.1 provides all possible transformations which preserve
the characteristic constants. Indeed, if A (z) = R(z+1)A (z)R−1 (z) then zeros
of det A (z) must be equal to those of det A (z) shifted by integers.
Proof. Let us prove the uniqueness of R first. Assume that there exist two
rational matrices R1 and R2 with needed properties. This means, in particular,
that the determinants of the matrices
−1
A(1) = R1 (z + 1)A(z)R1 (z)

and

−1
A(2) = R2 (z + 1)A(z)R2 (z)

vanish at the same set of mn points ai = ai + κi , none of which are different by
an integer. Denote by Y1r = R1 Y r and Y2r = R2 Y r the right canonical solutions
−1
of the corresponding equations. Then Y1r (Y2r )−1 = R1 R2 is a rational matrix
which tends to I at infinity. Moreover,
Y1r (Y2r )−1 (z + 1) = A(1) (z) Y1r (Y2r )−1 (z) A(2) (z)

−1

.

0, the equation above implies that
Since Y1r (Y2r )−1 is holomorphic for z
this function may only have poles at the points which are congruent to ai (zeros
of det A(2) (z)) and to the right of them. (Recall that two complex numbers
are congruent if their difference is an integer.) But since Y1r (Y2r )−1 is also
holomorphic for z
0, the same equation rewritten as
Y1r (Y2r )−1 (z) = A(1) (z)

−1

Y1r (Y2r )−1 (z + 1) A(2) (z)

implies that this function may only have poles at the points ai (zeros of
det A(1) (z)) or at the points congruent to them and to the left of them. Thus,
−1
Y1r (Y2r )−1 = R1 R2 is entire, and by Liouville’s theorem it is identically equal
to I. The proof of uniqueness is complete.
To prove the existence we note, first of all, that it suffices to provide a
proof if one of the κi ’s is equal to ±1 and one of the δj ’s is equal to ∓1 with
all other κ’s and δ’s equal to zero. The proof will consist of several steps.
Lemma 2.3. Let A(z) be an m by m matrix -valued function holomorphic
near z = a, and det A(z) = c(z − a) + O (z − a)2 as z → a, where c = 0.
Then there exists a unique (up to a constant) nonzero vector v ∈ Cm such that
A(a)v = 0. Furthermore, if B(z) is another matrix -valued function which is

holomorphic near z = a, then (BA−1 )(z) is holomorphic near z = a if and
only if B(a)v = 0.
Proof. Let us denote by E1 the matrix unit which has 1 as its (1, 1)-entry
and 0 as all other entries. Since det A(a) = 0, there exists a nondegenerate
constant matrix C such that A(a)CE1 = 0 (the first column of C must be a
0-eigenvector of A(a)). This implies that
H(z) = A(z)C(E1 (z − a)−1 + I − E1 )


1154

ALEXEI BORODIN

is holomorphic near z = a. On the other hand, det H(a) = c det C = 0.
Thus, A(a) = H(a)(I − E1 )C −1 annihilates a vector v if and only if C −1 v is
proportional to (1, 0, . . . , 0)t . Hence, v must be proportional to the first column
of C. The proof of the first part of the lemma is complete.
To prove the second part, we notice that
(BA−1 )(z) = B(z)C(E1 (z − a)−1 + I − E1 )H −1 (z)
which is bounded at z = a if and only if B(a)CE1 = 0.
More generally, we will denote by Ei the matrix unit defined by
1, k = l = i,
0, otherwise.

(Ei )kl =

Lemma 2.4 ([JM, §2 and Appendix A]). For any nonzero vector v =
(v1 , . . . , vm )t , Q ∈ Mat(m, C), a ∈ C, and i ∈ {1, . . . , m}, there exists a linear
matrix -valued function R(z) = R−1 (z − a) + R0 with the properties
R(z) I + Qz −1 + O z −2

z −Ei = I + O(z −1 ),

z → ∞,

R0 v = 0,
if and only if vi = 0. In this case,

R±1 (z)

is given by

R−1 (z) = I − Ei + R1 (z − a)−1 , det R(z) = z − a,
 −1
 vi
s=i Qis vs , k = l = i,


k = i, l = i,
−Qil ,
(R0 )kl =
−1
 −vi vk ,
k = i, l = i,


k = i, l = i,
δkl ,

R(z) = Ei (z − a) + R0 ,

−1
R1 = vi v Qi1 , . . . , Qi,i−1 , 1, Qi,i+1 , . . . , Qim .

The proof is straightforward.
Now we return to the proof of Theorem 2.1. Assume that κ1 = −1, δi = 1
for some i = 1, . . . , m, and all other κ’s and δ’s are zero. Since a1 is a simple
root of det A(z), by Lemma 2.3 there exists a unique (up to a constant) vector
v such that A(a)v = 0. Clearly, the condition vi = 0 defines a nonempty
Zariski open subset of M(a1 , . . . , amn ; δ1 , . . . ; δm ). On this subset, let us take
ˆ
R(z) to be the matrix afforded by Lemma 2.4 with a = a1 and Q = Y1 (we
ˆ0 = I, see Proposition 1.1). Then by the second part of Lemma
assume that Y
2.3, (A(z)R−1 (z))−1 = R(z)A−1 (z) is holomorphic and invertible near z = a1
(the invertibility follows from the fact that det R(z)A−1 (z) tends to a nonzero
value as z → a1 ). Thus, A(z) = R(z + 1)A(z)R−1 (z) is entire, hence, it is a
polynomial. Since
z + 1 − a1
det A(z) =
det A(z) = c (z + 1 − a1 )(z − a2 ) · · · (z − amn ), c = 0,
z − a1


1155

ISOMONODROMY TRANSFORMATIONS

the degree of A(z) is ≥ n. Looking at the asymptotics at infinity, we see that
deg A(z) ≤ n, which means that A is a polynomial of degree n:

A(z) = A0 z n + · · · + An ,

A0 = 0.

Denote by Y l,r the left and right canonical solutions of Y (z + 1) =
A(z)Y (z) (see Theorem 1.3 above). Then Y l,r := R Y l,r are solutions of
Y (z + 1) = A(z)Y (z). Moreover, their asymptotics at infinity at any left
(right) half-plane, by Lemma 2.4, is given by an expansion of the form (1.2)
ˆ
with Y0 = I, ρk = ρk for all k = 1, . . . , m, and
dk =

dk + 1, k = i,
dk ,
k = i.

This implies that A0 = diag(ρ1 , . . . , ρm ), and that Y l,r are the left and right
canonical solutions of the equation Y (z + 1) = A(z)Y (z). Indeed, their asymptotic expansion at infinity must also be a formal solution of the equation, the
fact that Y l,r are holomorphic for z
0 ( 0) follows from the analogous
property for Y l,r , and the location of possible poles of Y r is easily determined
from the equation.
For future reference let us also find a (unique up to a constant) vector v
such that At (a1 −1) v = 0. This means that R−t (a1 −1)At (a1 −1)Rt (a1 ) v = 0.
Lemma 2.4 then implies that
ˆ
ˆ
ˆ
ˆ
v = (Y1 )i1 , . . . , (Y1 )i,i−1 , 1, (Y1 )i,i+1 , . . . , (Y1 )im

t

is a solution. Note that vi = 0.
Now let us assume that κ1 = 1 and δi = −1 for some i = 1, . . . , m. By
Lemma 2.3, there exists a unique (up a to a constant) vector w such that
At (a1 )w = 0. The condition wi = 0 defines a nonempty Zariski open subset of
M(a1 , . . . , amn ; δ1 , . . . δm ). On this subset, denote by R (z) the rational matrixˆ
valued function afforded by Lemma 2.4 with a = a1 , v = w, and Q = −Y1t
ˆ0 = I). Set
(again, we assume that Y
−t

R(z) := (R ) (z − 1).
Then by Lemma 2.4
ˆ
R(z) I + Y1 z −1 + O z −2

z Ei = I + O(z −1 ),

z → ∞.

Furthermore, by Lemma 2.3, R−t (z + 1)A−t (z) is holomorphic and invertible
near z = a1 . Hence, A(z) = R(z + 1)A(z)R−1 (z) is entire (note that R−1 (z) =
(R )t (z − 1) is linear in z). The rest of the argument is similar to the case
κ1 = −1, δi = 1 considered above.


1156

ALEXEI BORODIN

Finding a solution w to A(a1 + 1)w = 0 is equivalent to finding a solution
to R (a1 )w = 0. One such solution has the form
ˆ
ˆ
ˆ
ˆ
w = −(Y1 )1i , . . . , −(Y1 )i−1,i , 1, −(Y1 )i+1,i , . . . , −(Y1 )mi

t

and all others are proportional to it. Note that its ith coordinate is nonzero.
From what was said above, it is obvious that the image of the map
M(a1 , . . . , amn ; δ1 , . . . , δm ) → M(a1 − 1, . . . , amn ; δ1 , . . . , δi + 1, . . . , δm )
is in the domain of definition of the map
M(a1 − 1, . . . , amn ; δ1 , . . . , δi + 1, . . . , δm ) → M(a1 , . . . , amn ; δ1 , . . . , δm )
and the other way around. On the other hand, the composition of these maps in
either order must be equal to the identity map due to the uniqueness argument
in the beginning of the proof. Hence, these maps are inverse to each other, and
they establish a bijection between their domains of definition. The rationality
of the maps follows from the explicit formula for R(z) in Lemma 2.4. The
proof of Theorem 2.1 is complete.
Remark 2.5. Quite similarly to Lemma 2.4, the multiplier R(z) can be
computed in the cases when two κ’s are equal to ±1 or two δ’s are equal to ±1
with all other κ’s and δ’s being zero; cf. [JM].
Assume κi = −1 and κj = 1. Denote by v and w the solutions of A(ai ) v
= 0 and At (aj ) w = 0. Then R exists if and only if (v, w) := v t w = wt v = 0,
in which case
R0

R0
z − ai
R(z) = I +
, det R(z) =
, R−1 (z) = I −
,
z − aj − 1
z − ai
z − aj − 1
aj − ai + 1
vwt .
R0 =
(v, w)
Now assume δi = 1, δj = −1. Then we must have det R(z) = 1 and
ˆ
ˆ
R(z) I + Y1 z −1 + Y2 z −2 + O(z −3 ) z Ej −Ei = I + O(z −1 ),

z → ∞.

ˆ
The solution exists if and only if (Y1 )ij = 0, in which case it has the form
−1
R−1 (z) = Ej z + R0 ,

R(z) = Ei z + R0 ,
with (R0 )kl given by
l=i
k=i
k=j

k = i, j

ˆ
−(Y2 )ij +

s=i

ˆ
ˆ
(Y1 )is (Y1 )sj

ˆ
(Y1 )ij
1
ˆ
(Y1 )ij
ˆ
(Y1 )
− ˆ kj
(Y1 )
ij

l=j
ˆ
−(Y1 )

ij

l = i, j
ˆ

−(Y1 )

il

0

0

0

δkl ,


1157

ISOMONODROMY TRANSFORMATIONS
−1
and (R0 )kl given by

l=i
0

k=i
k=j

−

k = i, j

1

ˆ
(Y1 )ij

l=j
ˆ
(Y1 )ij

−

ˆ
(Y2 )ij
ˆ
(Y1 )

0

ij

ˆ
+ (Y1 )jj

ˆ
(Y1 )kj

l = i, j
0
−

ˆ
(Y1 )il

ˆ
(Y1 )ij

δkl .

3. Difference Schlesinger equations
In this section we give a different description for the transformations
A → A of Theorem 2.1 with
κi1 = · · · = κim = ±1,

δ1 = · · · = δm = ∓1,

and all other κi ’s equal to zero, and for compositions of such transformations.
In what follows we always assume that our matrix polynomials A(z) =
A0 z n + . . . have nondegenerate highest coefficients: det A0 = 0. We also
assume that mn roots of the equation det A(z) = 0 are pairwise distinct; we
will call them the eigenvalues of A(z). For an eigenvalue a, there exists a
(unique) nonzero vector v such that A(a) v = 0, see Lemma 2.3. We will
call v the eigenvector of A(z) corresponding to the eigenvalue a. The word
generic everywhere below stands for “belonging to a Zariski open subset” of
the corresponding algebraic variety.
We start with few simple preliminary lemmas.
Lemma 3.1. The sets of eigenvalues and corresponding eigenvectors define A(z) up to multiplication by a constant nondegenerate matrix on the left.
Proof. If there are two matrix polynomials A and A with the same eigenvalues and eigenvectors, then (A (z))−1 A (z) has no singularities in the finite
plane. Moreover, since the degrees of A (z) and A (z) are equal, (A (z))−1 A (z)
∼ (A0 )−1 A0 as z → ∞. Liouville’s theorem concludes the proof.
We will say that z − B, B ∈ Mat(m, C), is a right divisor of A(z) if
ˆ
ˆ
A(z) = A(z)(z − B), where A(z) is a polynomial of degree n − 1.

Lemma 3.2. A linear function z − B is a right divisor of A(z) if and only
if
A0 B n + A1 B n−1 + · · · + An = 0.
Proof. See, e.g., [GLR].


1158

ALEXEI BORODIN

Lemma 3.3. Let α1 , . . . , αm be eigenvalues of A(z) and v1 , . . . , vm be the
corresponding eigenvectors. Assume that v1 , . . . , vm are linearly independent.
Take B ∈ Mat(m, C) such that Bvi = αi vi , i = 1, . . . , m. Then z − B is a right
divisor of A(z). Moreover, B is uniquely defined by the conditions that z − B
is a right divisor of A(z) and Sp(B) = {α1 , . . . , αm }.
Proof. For all i = 1, . . . , m,
n−1
n
(A0 B n + A1 B n−1 + · · · + An )vi = (A0 αi + A1 αi + · · · + An )vi = A(αi )vi = 0.

Lemma 3.2 shows that z − B is a right divisor of A(z).
To show uniqueness, assume that
ˆ
ˆ
A(z) = A (z)(z − B ) = A (z)(z − B ).
This implies (A (z))−1 A (z) = (z − B )(z − B )−1 . Possible singularities of
the right-hand side of this equality are z = αi , i = 1, . . . , m, while possible singularities of the left-hand side are all other eigenvalues of A(z). Since
the eigenvalues of A(z) are pairwise distinct, both sides are entire. But
(z − B )(z − B )−1 tends to I as z → ∞. Hence, by Liouville’s theorem,
B =B .

Now let us assume that the eigenvalues a1 , . . . , amn of A(z) are divided
into n groups of m numbers:
(1)

(n)

{a1 , . . . , amn } = {a1 , . . . , a(1) } ∪ · · · ∪ {a1 , . . . , a(n) }.
m
m
Lemma 3.3 shows that for a generic A(z) we can construct uniquely
defined B1 , . . . , Bn ∈ Mat(m, C) such that for any i = 1, . . . , n, Sp(Bi ) =
(i)
(i)
{a1 , . . . , am } and z −Bi is a right divisor of A(z).4 By Lemma 3.1, B1 , . . . , Bn
define A(z) uniquely up to a left constant factor, because the eigenvectors of
Bi must be eigenvectors of A(z).
(i)

Lemma 3.4. For generic B1 , . . . , Bn ∈ Mat(m, C) with Sp(Bi ) = {aj },
there exists a unique monic degree n polynomial A(z) = z n + A1 z n−1 + . . .
such that z − Bi are its right divisors. The matrix elements of A1 , . . . , An are
(i)
rational functions of the matrix elements of B1 , . . . , Bn and eigenvalues aj .
Remark 3.5. 1. Later on we will show that, in fact, these rational func(i)
tions do not depend on aj .
2. Clearly, the condition of A(z) being monic can be replaced by the
condition of A(z) having a prescribed nondegenerate highest coefficient A0 .
4

It is obvious that the condition on A(z), used in Lemma 3.3, is an open condition. The

(k)
corresponding set is nonempty because it contains diagonal A(z) where the ai
are the
roots of Akk (z). Similar remarks apply to all appearances of the word “generic” below.


1159

ISOMONODROMY TRANSFORMATIONS

Proof. The uniqueness follows from Lemma 3.1. To prove the existence
part, we use induction on n. For n = 1 the claim is obvious. Assume that we
ˆ
ˆ
have already constructed A(z) = z n−1 + A1 z n−1 + . . . such that B1 , . . . , Bn−1
are its right divisors. Let {vi } be the eigenvectors of Bn with eigenvalues
(n)
(n)
ˆ (n)
{ai }. Set wi = A(ai )vi and take X ∈ Mat(m, C) such that Xwi = ai wi
for all i = 1, . . . , m. (The vectors {wi } are linearly independent generically.)
ˆ
Then A(z) = (z − X)A(z) has all needed properties. Indeed, we just need to
check that z − Bn is its right divisor (the rationality follows from the fact that
computing the eigenvectors with known eigenvalues is a rational operation).
For any i = 1, . . . , m,
(n) n

(n) n−1

n
n−1
(Bn + A1 Bn + · · · + An )vi = (ai ) + A1 (ai )
(n)

= (ai

(n)

+ · · · + An vi
(n)

ˆ
− X)A(ai )vi = (ai

− X)wi = 0.

Lemma 3.2 concludes the proof.
Thus, we have a birational map between matrix polynomials A(z) =
A0 z n + . . . with a fixed nondegenerate highest coefficient and fixed mutually distinct eigenvalues divided into n groups of m numbers each, and sets of
right divisors {z − B1 , . . . , z − Bn } with Bi having the eigenvalues from the ith
group. We will treat {Bi } as a different set of coordinates for A(z).
It turns out that in these coordinates some multipliers R(z) of Theorem 2.1
(i)
take a very simple form. We will redenote by κj the numbers κ1 , . . . , κmn used
in Theorem 2.1 in accordance with our new notation for the eigenvalues of A(z).
Denote the transformation of Theorem 2.1 with
n
(i)
κj

= −ki ∈ Z,

i = 1, . . . , n, j = 1, . . . , m;

δ1 = · · · = δm =

ki
i=1

by S(k1 , . . . , kn ).
(i)

Proposition 3.6. The multiplier R(z) for S(0, . . . , 0, 1 , 0, . . . , 0) is equal
(i)
(i)
to the right divisor z −Bi of A(z) corresponding to the eigenvalues a1 , . . . , am .
(i)

(i)

Proof. It is easy to see that if Bi has eigenvalues a1 , . . . , am , and z − Bi
is a right divisor of A(z) then R(z) = z − Bi satisfies all the conditions of
Theorem 2.1.
Conversely, if R(z) is the corresponding multiplier then R(z) is a product
of n elementary multipliers with one κ equal to −1 and one δ equal to +1.
The explicit construction of the proof of Theorem 2.1 shows that all these
multipliers are polynomials; hence, R(z) is a polynomial. The fact that δ1 =
· · · = δm implies that R(z) is a linear polynomial of the form z−B for some B ∈

1160

ALEXEI BORODIN

Mat(m, C) (to see this, it suffices to look at the asymptotics of the canonical
solutions). We have
A(z) = R−1 (z + 1)A(z)R(z) = (z + I − B)−1 A(z)(z − B).
Comparing the determinants of both sides we conclude that Sp(B) =
(i)
(i)
{a1 , . . . , am }. Since no two eigenvalues are different by an integer, B and
B − I have no common eigenvalues. This implies that (z + I − B)−1 A(z) must
be a polynomial, and hence z − B is a right divisor of A(z).
For any k = (k1 , . . . , kn ) ∈ Zn we introduce matrices B1 (k), . . . , Bn (k)
such that the right divisors of S(k1 , . . . , kn )A(z) have the form z − Bi (k) with
(i)

Sp(Bi (k)) = {a1 − ki , . . . , a(i) − ki },
n

i = 1, . . . , n.

They are defined for generic A(z) from the varieties M(· · · ) introduced in the
previous section.
Proposition 3.7 (difference Schlesinger equations). The matrices {Bi (k)}
(whenever they exist) satisfy the following equations:
(3.1)

Bi (. . . ) − Bi (. . . , kj + 1, . . . ) = Bj (. . . ) − Bj (. . . , ki + 1, . . . ),

(3.2)

Bj (. . . , ki + 1, . . . )Bi (. . . ) = Bi (. . . , kj + 1, . . . )Bj (. . . ),

(3.3)

Bi (k1 + 1, . . . , kn + 1) = A−1 Bi (k1 , . . . , kn )A0 − I,
0

where i, j = 1, . . . , n, and dots in the arguments mean that other kl ’s remain
unchanged.
Remark 3.8. The first two equations above are equivalent to
(3.4)

z − Bi (. . . , kj + 1, . . . ) z − Bj (. . . )
= z − Bj (. . . , ki + 1, . . . ) z − Bi (. . . ) .

Proof of Proposition 3.7.

The uniqueness part of Theorem 2.1 implies

that
(i)

(j)

S(0, . . . , 0, 1 , 0, . . . , 0) ◦ S(0, . . . , 0, 1 , 0, . . . , 0) ◦ S(k1 , . . . , kn )
(j)

(i)

= S(0, . . . , 0, 1 , 0, . . . , 0) ◦ S(0, . . . , 0, 1 , 0, . . . , 0) ◦ S(k1 , . . . , kn ).
Thus, the corresponding products of the multipliers are equal, which gives
(3.4). This proves (3.1), (3.2). The relation (3.3) follows from the fact that
the multiplier for S(1, . . . , 1) is equal to A−1 A(z), and A(z) = S(1, . . . , 1) =
0
A−1 A(z + 1)A0 . This means that the right divisors for A(z) can be obtained
0
from those for A(z) by shifting z by 1 and conjugating by A0 .


1161

ISOMONODROMY TRANSFORMATIONS
(i) n,m

Theorem 3.9. Fix mn complex numbers aj i=1,j=1 such that no two
of them are different by an integer, and an integer M > 0. Then for generic
(i) m
B1 , . . . , Bn ∈ Mat(m, C), Sp(Bi ) = aj j=1 , there exists a unique solution
{Bi (k1 , . . . , kn ) : max |ki | ≤ M }
i=1,...,n

of the difference Schlesinger equations (3.1)–(3.3) with
Sp(A0 ) = {ρ1 , . . . , ρn },

ρi /ρj ∈ R for i = j,
/

ρi = 0 for i = 1, . . . , n,

such that
Sp(Bi (k1 , . . . , kn )) = Sp(Bi ) − ki and Bi (0, . . . , 0) = Bi

for all i = 1, . . . , n.

The matrix elements of Bi (k) are rational functions of the matrix elements
of the initial conditions {Bi }n . Moreover, these rational functions do not
i=1
(j)
depend on the eigenvalues ai .
Remark 3.10. As we will see later, this theorem also extends to the case
of arbitrary invertible A0 .
Proof. The existence and rationality of the flows have already been proved.
Indeed, without loss of generality we can assume that A0 is diagonal (the equations (3.1)–(3.3) remain intact if we conjugate all Bi (k) and A0 by the same
constant matrix). By Lemma 3.4 we can construct a (unique) degree n polynomial A(z) with the highest coefficient A0 , such that {Bi } is the set of its right
divisors. Then, using Theorem 2.1, we can define S(k) and hence {Bi (k)}. By
Proposition 3.7 they will satisfy (3.1)–(3.3). Moreover, all operations involved
in this construction are rational.
Thus, it remains to prove uniqueness and the fact that the rational functions involved do not depend on the eigenvalues. A simple computation shows
that for any X, Y, S, T ∈ Mat(m, C), the relation (z−X)(z−Y ) = (z−S)(z−T )
implies
(3.5)

Y = (X − S)−1 S(X − S),

T = (X − S)−1 X(X − S),

(3.6)

X = (Y − T )T (Y − T )−1 ,

S = (Y − T )Y (Y − T )−1 ,

whenever the corresponding matrices are invertible; cf. [GRW]. Applying this
observation to (3.4), we see that, generically, {Bi = Bi (0, . . . , 0)} uniquely
define all
(3.7)

(i)

(i)

(i)

(i)

Bi (ε1 , . . . , εi−1 , 0 , εi+1 , . . . , ε(i) ),
n

(i)

εj = 0, 1.

Moreover, they are all given by rational expressions involving the initial conditions {Bi } only. To move further, we need the following lemma.


1162

ALEXEI BORODIN

Lemma 3.11. For generic X, Y ∈ Mat(m, C) with fixed disjoint spectra,
there exist unique S, T ∈ Mat(m, C) such that
(z − X)(z − Y ) = (z − S)(z − T ),

Sp(S) = Sp(Y ),

Sp(T ) = Sp(X).

The matrix elements of S and T are rational functions of the matrix elements
of X and Y which do not depend on the spectra of X and Y .
Proof 1.
Lemma 3.3 proves the uniqueness and shows how to construct T if we know the eigenvalues x1 , . . . , xm of X and vectors vi such that
(xi − X)(xi − Y )vi = 0. If we normalize vi ’s in the same way, for example, by
requiring the first coordinate to be equal to 1 (this can be done generically),
then using the construction of Lemma 3.3 we obtain the matrix elements of T
as rational functions in the matrix elements of X, Y and x1 , . . . , xn . However,
it is easy to see that these rational functions are symmetric with respect to
the permutations of x1 , . . . , xn , which means that they depend only on the
elementary symmetric functions i1 <···coefficients of the characteristic polynomial of X, and hence they are expressible as polynomials in the matrix elements of X.
Proof 2 (see [O]). The uniqueness follows from Lemma 3.3. To prove the
existence, denote by Λ the solution of the equation Y Λ − ΛX = I. Generically,
it exists, it is unique and invertible. Set
S = X + Λ−1 ,

T = Y − Λ−1 .

Then it is easy to see that (z − X)(z − Y ) = (z − S)(z − T ). Furthermore, if Y

and T have a common eigenvalue then they must have a common eigenvector,
which contradicts the invertibility of Y − T = Λ−1 . Hence, Sp(T ) = Sp(X)
and Sp(S) = Sp(Y ).
Remark 3.12. In the case of 2 by 2 matrices, it is not hard to produce an
explicit formula for S and T in terms of X and Y :
(3.8)

S = (X + Y − Tr Y )Y (X + Y − Tr Y )−1 ,
T = (X + Y − Tr X)−1 X(X + Y − Tr X).

Now let us return to the proof of Theorem 3.9. Recall that we already
proved that the initial conditions define (3.7) uniquely. Now let us use (3.4)
with
(j)

(k1 , . . . , kn ) = (1, . . . , 1, 0 , 1, . . . , 1),

j = i.

By (3.3), we know what Bi (1, . . . , 1) is. Thus, we know both matrices on
the left-hand side of (3.4), and hence, by Lemma 3.11, we can compute both
(j)

matrices on the right-hand side of (3.4), in particular, Bi (1, . . . , 1, 0 , 1, . . . , 1).


1163

ISOMONODROMY TRANSFORMATIONS

Now take (3.4) with
(j)

(l)

(k1 , . . . , kn ) = (1, . . . , 1, 0 , 1, . . . , 1, 0 , 1, . . . , 1),
where i, j, l are pairwise distinct. Applying Lemma 3.11 again, we find all
(j)

(l)

Bi (1, . . . , 1, 0 , 1, . . . , 1, 0 , 1, . . . , 1).
Continuing the computations in this fashion (changing one more 1 to 0 in
(k1 , . . . , kn ) on each step), we obtain all
(i)

(i)

(i)

(i)

(i)

Bi (ε1 , . . . , εi−1 , 1 , εi+1 , . . . , ε(i) ),
n

εj = 0, 1.

Together with (3.3) (and (3.7)) this computes all Bi (k) with max |ki | ≤ 1.

Iterating this procedure, we complete the proof.
4. An alternative description of the Schlesinger flows
The goal of this section is to provide yet another set of coordinates for the
polynomials A(z), in which the flows described in the previous section can be
easily defined. In particular, this will lead to a different proof of Theorem 3.9,
which will be valid for an arbitrary invertible A0 .
Proposition 4.1. With the assumptions of Theorem 3.9, the monic degree n polynomial
(z − B1 (0, 1, . . . , 1))(z − B2 (0, 0, 1, . . . , 1)) · · · (z − Bn (0, . . . , 0))
has z − Bi , i = 1, . . . , n, as its right divisors.
This statement and Theorem 3.9 provide a proof for Remark 3.5(1).
Proof. Using (3.4) we obtain, for (j > i),
(j)

z − Bi (0, . . . , 0, 1 , . . . , 1)
(i)

=

(j+1)

z − Bj (0, . . . , 0, 1 , . . . , 1)
(j+1)

z − Bj (0, . . . , 0, 1 , 0, . . . , 0, 1 , . . . , 1)

(j+1)

z − Bi (0, . . . , 0, 1 , . . . , 1) .

Using this commutation relation, we can move the factor (z − Bi (· · · )), in the

product above, to the right most position, where it will turn into
(z − Bi (0, . . . , 0)) = (z − Bi ).
Let us introduce the notation (l1 , . . . , ln ∈ Z)
Ci (l1 , . . . , ln ) := Bi (l1 , . . . , li , li+1 + 1, . . . , ln + 1),

i = 1, . . . , n.


1164

ALEXEI BORODIN

If we denote by A(z) the polynomial of degree n with highest coefficient A0
such that the {z − Bi } are its left divisors, then the definition of Bi (k) and
Proposition 4.1 imply that for any l = (l1 , . . . , ln ) ∈ Zn ,
S(l1 , . . . , ln )A(z) = A0 z − C1 (l) · · · z − Cn (l) .

(4.1)

(To apply Proposition 4.1, we also used an easy fact that for any solution
{Bi (k)} of (3.1)–(3.3) and any l1 , . . . , ln ∈ Z,
Bi (k1 , . . . , kn ) := Bi (k1 + l1 , . . . , kn + ln ),

i = 1, . . . , n,

also form a solution of (3.1)–(3.3).)
Lemma 4.2. The map {Bi } → {Ci } is birational.
Proof. The rationality of the forward map follows from Theorem 3.9.
The rationality of the inverse map follows from Lemma 3.3 (indeed, we just
need to find the right divisors of the known matrix S(l1 , . . . , ln )A(z)). Even

though it looks like to construct Bi we need to know the eigenvalues of Ci , it
is clear that by normalizing the eigenvectors of A(z) corresponding to these
eigenvalues, in the same way, we will obtain a formula for Bi which will be
symmetric with respect to the permutations of these eigenvalues. Thus we can
rewrite it through the matrix elements of Ci ’s only (this argument was used in
the first proof of Lemma 3.11 above).
Our goal is to describe the transformations S(k) in terms of {Ci }. We
need a preliminary lemma which generalizes Lemma 3.11.
Lemma 4.3. For generic X1 , . . . , XN ∈ Mat(m, C) with fixed disjoint
spectra and any permutation σ ∈ SN , there exist unique Y1 , . . . , YN ∈ Mat(m, C)
such that Sp(Yi ) = Sp(Xi ) for all i = 1, . . . , N , and
(z − X1 ) · · · (z − XN ) = (z − Yσ(1) ) · · · (z − Yσ(N ) ).
The matrix elements of {Yi } are rational functions of the matrix elements of
{Xi } which do not depend on the spectra of {Xi }.
Proof. The existence and rationality claims follow from Lemma 3.11,
because elementary transpositions (i, i + 1) generate the symmetric group SN .
To show uniqueness, we rewrite the equality
(z − Y1 ) · · · (z − YN ) = (z − Y1 ) · · · (z − YN ),

Sp(Yi ) = Sp(Yi ),

in the form
(z − Y1 )−1 (z − Y1 ) = (z − Y2 ) · · · (z − YN )

(z − Y2 ) · · · (z − YN )

−1

.

If the spectrum of Y1 is disjoint with the spectra of Y2 , . . . , Ym , then both sides
of the last equality are entire because they cannot possibly have common poles.